Solving Einstein's field equations

1
Solving Einstein's field equations
for space-times with symmetries
Integrability structures and
nonlinear dynamics of
interacting fields
G.Alekseev
Many languages of integrability
Introduction
Gravitational and electromagnetic solitons
Lecture 1
Monodromy transform approach
Lecture 2
Solutions for black holes in the external fields
Solution of the characteristic initial value
problem Colliding gravitational and
electromagnetic waves
Lecture 3
2
Integrability of Einsteins equations with
symmetries
Integrable cases - Vacuum
gravitational fields
- Einstein Maxwell - Weyl fields
- Ideal
fluid with
- some string gravity models
mathematical context - infinite hierarchies
of exact solutions,
- initial and boundary value
problems,
- asymptotical behaviour
physical context - supeposition of
stat. axisymm. fields,
- nonlinear interacting waves,
-
inhomogeneous cosmological models
3
Many "languages" of integrability

• Associated linear systems and spectral
  problems
• Infinite-dimensional algebra of internal
  symmetries
• Solution generating procedures (arbitrary seed)
• -- Solitons,
• -- Backlund transformations,
• -- Symmetry transformations
• Infinite hierarchies of exact solutions
• -- Meromorfic on the Riemann sphere
• -- Meromorfic on the Riemann surfaces
  (finite gap solutions)
• Prolongation structures
• Geroch conjecture
• Riemann Hielbert and Homogeneous Hilbert
  problems,
• Various linear singular integral equation
  methods
• Initial and boundary value problems
• -- Characteristic initial value problems
• -- Boundary value problems for
  stationary axisymmetric fields
• Twistor theory of the Ernst equation

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Integrable reductions of the Einstein's field
equations
Gravitational fields in vacuum
Elektrovacuum Einstein - Maxwell fields
Gravity model with axion, dilaton and E-H fields
Bosonic sector of heterotic string effective
action
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1)
Space-times with one Killing vector
Einstein Maxwell fields the Ernst-like
equations
SU(2,1) symmetric form of dynamical equations
1)
W.Kinnersley, J. Math.Phys. (1973)
6
1)
Space-times with two commuting Killing vectors
Isometry group with 2-surface orthogonal orbits
The Einsteins field equations
-- the constraint equations
-- the dynamical equations
-- the dynamical equations
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Space-times with two commuting Killing vectors
Generalized Weyl coordinates
Geometrically defined coordinates
8
Lecture 1
Integrable reductions of Einstein equations
Belinski Zakharov vacuum solitons
Einstein Maxwell solitons
Examples of soliton solutions
9
Ernst vacuum equation
Kinnersley self-dual form of the reduced vacuum
equations
Belinski Zakharov form of reduced vacuum
equations
2x2-matrix form of self-dual reduced vacuum
equations
10
1)
Belinski - Zakharov inverse scattering approach
Dynamical equations for vacuum
Associated spectral problem
Dressing method for constructing solutions
V.Belinski V.Zakharov,, JETP 1978 1979
1)
11
1)
Formulation of the matrix Riemann problem
Riemann problem for dressing matrix
Linear singular integral equations
Constraints for dressing matrix
V.Belinski V.Zakharov,, JETP 1978 1979
1)
12
1)
( - solitons)
Vacuum solitons
Soliton ansatz for dressing matrix
V.Belinski V.Zakharov,, JETP 1978 1979
1)
13
1)
Determinant form of Belinski Zakharov vacuum
solitons
Stationary axisymmetric solitons on the Minkowski
background
a set of 4 N arbitrary real or pairwise complex
conjugated constants
1)
GA, Sov.Phys.Dokl. (1981)
14
Integrable reductions of Einstein-Maxwell
equations
Spacetime metric and electromagnetic potential
15
Electrovacuum fields
Ernst potentials
Ernst equations
16
3x3-matrix form of Einstein Maxwell equaations
17
1)
Spectral problem for Einstein Maxwell fields
For vacuum
1)
GA, JETP Lett.. (1980) Proc. Steklov Inst.
Math. (1988) Physica D. (1999)
18
1)
(w - solitons)
Einstein - Maxwell solitons
Dressing matrix
Soliton ansatz for dressing matrix
--- a set of 3 N arbitrary complex constants
1)
GA, JETP Lett. (1980) Proc. Steklov Inst. Math.
(1988) Physica D. (1999)
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Example one-soliton solution on the Minkowski
background
-- Superextreme part of the Kerr-Newman
solution
-- mass -- NUT-parameter -- angular momentum --
electric charge -- magnetic charge
1)
Two-soliton solution on the Minkowski background
-- Interaction of two superextreme
Kerr-Newman sources
1)
GA, Proc. Steklov Inst. Math. (1988) Physica D.
(1999)
20
Two-soliton solution on the Minkowski background
-- Interaction of two superextreme
Kerr-Newman sources
21
Soliton gravitational and electromagnetic waves
on the Minkowski background